Question

Let $P(3, 2, 3)$, $Q(4, 6, 2)$, and $R(7, 3, 2)$ be the vertices of $\triangle PQR$. Then, the angle $\angle QPR$ is

• A. $\frac{\pi}{6}$
• B. $\cos^{-1} \left( \frac{7}{18} \right)$
• C. $\cos^{-1} \left( \frac{1}{18} \right)$
• D. $\frac{\pi}{3}$

Answer 1

Answer: (D)

$\frac{\pi}{3}$

Answer 2

Solution: To find the angle ∠QPR, we calculate the direction ratios of PR and PQ.

Step 1. Direction Ratio of PR:
PR = (7 − 3, 3 − 2, 2 − 3) = (4, 1, −1)

Step 2. Direction Ratio of PQ:
PQ = (4 − 3, 6 − 2, 2 − 3) = (1, 4, −1)

Step 3. Calculating cosθ:
$\cosθ = \frac{4·1 + 1·4 + (−1)·(−1)}{\sqrt{18}·\sqrt{18}} = \frac{4 + 4 + 1}{18} = \frac{9}{18} = \frac{1}{2}$
Step 4. Therefore:

$θ = \cos⁻¹\left(\frac{1}{2}\right) = \frac{π}{3}$

The Correct Answer is:$\frac{π}{3}$